![]() Now, you may ask is it possible to group different terms into a family and still get the same final answer. So we factor it out and that leaves us with our final answer: With that in mind, we can factor this entire polynomial by recognizing there is a common factor between the two terms: ( x − 7 ) (x-7) ( x − 7 ). ![]() What is important now is to consider each of our "groups" from before as their own terms in the polynomial, with x 2 ( x − 7 ) x^2(x-7) x 2 ( x − 7 ) being the first term, and + 2 ( x − 7 ) +2(x-7) + 2 ( x − 7 ) being the second term. This gives us a more complicated looking polynomial that is: So we take 2 out of the family and we end up with:Īfter factoring each group individually, we now need to put the groups together. Going on to the next group which is ( + 2 x − 14 ) (+2x-14) ( + 2 x − 14 ), the common factor here is 2. After taking x 2 x^2 x 2 out, we end up with: In the group that is ( x 3 − 7 x 2 ) (x^3-7x^2) ( x 3 − 7 x 2 ), we can take out the common factor x 2 x^2 x 2 of this family. Now that we have done our grouping step, next we need to factor each of these groups using skills we've developed in the past. In this case, the groups we will make are: You will see later that this doesn't necessarily matter, but it is the easiest way to do it. Choosing what groups to make varies from problem to problem, but, in most cases, we are usually going to group the 2 highest powers together and then the lowest 2 or 1 powers together. This is the trickiest part of solving these kinds of problems. The best way to learn this technique is to do some factoring by grouping examples!įactor the following polynomial by grouping: Lastly, for a video explanation of all of this, see our video on how to factor by grouping. Remember, the degree of a polynomial just related to the highest power on the independent variable x. The process is the same for any degree of polynomial, whether we be factoring quadratics by grouping, cubics by grouping, and beyond. Once we break it up into groups, we can factor using the methods we've learned from factoring quadratic and simpler polynomials. When we are factoring by grouping, all we are really doing is breaking up our polynomial into easier-to-factor groups or "families" so that we can better approach the problem. It is important to understand what we mean when we say "grouping". What makes factoring by grouping so powerful, however, is its ability to help us to factor higher degree polynomials like cubics with relative ease. Factoring polynomials by grouping is just another technique we can use, similar to others you've likely seen in the past. Now that we have a good understanding of what it means to factor in its most general terms, let's look at factoring by grouping. ![]() Not too complicated after all!Ĭheck out our videos covering how to find the greatest common factor of polynomials, factoring polynomials with common factor, as well as factoring trinomials with leading coefficient not 1. The factor of a polynomial is just a value of the independent value (usually x) that makes an entire polynomial equation to zero. When we're looking at factoring polynomials, in particular, the meaning of a factor isn't all that different. 7 is not a factor of 20 because, when 20 is divided by 7, we get 2.86, which is not a whole number (this is the same as saying 2 and a remainder of 0.86). In its simplest terms, consider the following: 4 is a factor of 20 because, when 20 is divided by 4, we get the whole number 5 and no remainder. With regards to division, a factor is just a term or expression that, when another term or expression is divided by this factor, the remainder is equal to zero. The understanding of what factors are is crucial to all of mathematics, and it is a term you will hear again and again as you progress with your studies. Before we get into the details of factoring polynomials by grouping, let's do a quick review of the general process of factoring itself.įirst, we need to know what exactly a "factor" is. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |